Speed and Velocity in Physics Problems

By Steven Holzner

In physics terms, what is speed? It’s the same as the conventional idea of speed: Speed is distance divided by time, which is what a speedometer measures. The related term velocity refers to a speed with an associated direction. To measure velocity, you might use a speedometer in combination with a compass. Sometimes, you are interested in the average velocity over a period of time instead of velocity at a particular instant. Therefore, we define the average velocity of an object as displacement (distance in a particular direction) divided by time.

For example, if you went a displacement s in a time t, then your average velocity, v, is determined as follows:

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Technically speaking, average velocity is the change in position divided by the change in time, so you also can represent it like this if, for example, you’re moving along the x axis:

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Sample question

  1. Suppose that you want to drive from New York City to Los Angeles to visit your uncle’s family, a distance of about 2,781 miles.

    The trip takes you four days. What was your average speed in miles per hour?

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    1. Start by figuring out your average speed (the distance traveled divided by the time taken to travel that distance):

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    2. Okay, the average speed is 695.25, but 695.25 what?

      This solution divides miles by days, so it’s 695.25 miles per day — not exactly a standard unit of measurement. So what is that in miles per hour? To determine that, you cancel “days” out of this equation and put in “hours.” Because 24 hours are in a day, you can multiply as follows (note that “days” cancel out, leaving miles over hours, or miles per hour):

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      So your speed was 28.97 miles per hour. That’s your average speed, averaged over both day and night.

Practice questions

  1. Suppose that you used your new SpeedPass to get you through the tollbooths at both ends of your trip, which was 90.0 miles on the turnpike and took you 1 hour and 15 minutes.

    On your return home, you’re surprised to find a traffic ticket for speeding in the mail. How fast did you go, on average, between the tollbooths? Was the turnpike authority justified in sending you a ticket, given that the speed limit was 65 mph?

  2. Suppose that you and a friend are determined to find out who drives faster.

    You both start your trips in Chicago. Driving nonstop, you reach Los Angeles — a distance of 2,018 miles — in 1.29 days, and your friend, also driving nonstop, reaches Miami — a distance of 1,380 miles — in 0.89 days. Who drove faster?

Following are answers to the practice questions:

  1. v = 72.0 miles an hour. The ticket was justified.

    It took you one hour and fifteen minutes, or 1.25 hours, to travel 90.0 miles.

    Divide 90.0 miles by 1.25 hours:

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    This is faster than the speed limit of 65 miles/hour.

  2. Your speed = 1,564 miles per day; your friend’s speed = 1,550 miles per day. You’re faster.

    Note that to simply compare speeds, there’s no need to convert to miles per hour — miles per day will do fine. First, calculate your speed:

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    Next, calculate your friend’s speed:

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    So you were faster than your friend.